Precision Engineering - Pedro P Balestrassi

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Precision Engineering

j ou rn a l h o m e pa g e :w w w . e l s e v i e r . c o m / l o c a t e / p r e c i s i o n

A normal boundary intersection approach to multiresponse robust optimization of the surface roughness in end milling process with combined arrays

T.G. Brito, A.P. Paiva, J.R. Ferreira, J.H.F. Gomes, P.P. Balestrassi

^∗

InstituteofIndustrialEngineering,FederalUniversityofItajubá,37500-903Itajuba,MinasGerais,Brazil

a r t i c l e i n f o

Articlehistory:

Received15April2013

Receivedinrevisedform11January2014 Accepted22February2014

Availableonline6March2014

Keywords:

Multipleobjectiveprogramming Robustparameterdesign(RPD) Normalboundaryintersection(NBI) Endmillingprocess

Surfaceroughness

a b s t r a c t

Robustparameterdesign(RPD)hasrecentlybeenappliedinmodernindustriesinalargedealofprocesses.

Thistechniqueisoccasionallyemployedasamultiobjectiveoptimizationapproachusingweightedsums asatrade-offstrategy;insuchcases,however,aconsiderablenumberofgapshavearisen.Inthispaper, theuseofnormalboundaryintersection(NBI)methodcoupledwithmean-squarederror(MSE)functions isproposed.ThisapproachiscapableofgeneratingequispacedParetofrontiersforabi-objectiverobust designmodel,independentoftherelativescalesoftheobjectivefunctions.Toverifytheadequacyofthis proposal,acentralcompositedesign(CCD)isdevelopedwithcombinedarraysfortheAISI1045steel endmillingprocess.Inthiscasestudy,aCCDwiththreenoisefactorsandfourcontrolfactorsareused tocreatethemeanandvarianceequationsforMSEoftwoqualitycharacteristics.Thenumericalresults indicatetheNBI-MSEapproachiscapableofgeneratingaconvexandequispacedParetofrontiertoMSE functionsofsurfaceroughness,thusnullifyingthedrawbacksofweightedsums.Moreover,theresults showthattheachievedoptimumlessensthesensitivityoftheendmillingprocesstothevariability transmittedbythenoisefactors.

©2014ElsevierInc.Allrightsreserved.

1. Introduction

Tomake aprocess lesssensitivetotheaction ofnoise vari- ables,researchershavedevelopedadesignofexperiments(DOE) approach that promotes thebest levelsof control factors. The approach,knownasrobustparameterdesign(RPD),improvesthe variabilitycontroland minimizesthebias.Theways ofutilizing RPDcanvary.Forexample,intheirestimatingofcuttingcondi- tionsofsurfaceroughnessinendmillingmachiningprocesses[1], usedkernel-basedregressionandgeneticalgorithms(GA).Employ- ingahybridTaguchi-geneticlearningalgorithm[2],reliedonan adaptivenetwork-basedfuzzyinferencesystemtopredictsurface roughnessinendmillingprocesses.Tominimizesurfaceroughness inendmillingmachiningprocesses[3],studiedanapplicationofGA soastooptimizecuttingconditions.

∗Correspondingauthorat:AvBPS1303,37500-903Itajubá,MG,Brazil.

Tel.:+553536291150;fax:+553588776958.

E-mailaddresses:engtaarc.gb@ig.com.br(T.G.Brito), andersonppaiva@unifei.edu.br(A.P.Paiva),jorofe@unifei.edu.br

(J.R.Ferreira),zehenriquefg@yahoo.com.br(J.H.F.Gomes),pedro@unifei.edu.br, ppbalestrassi@gmail.com(P.P.Balestrassi).

ThisworkpresentsanRPDthatwillfacilitatetheadaptivecon- trol application in end milling processes as well as contribute to computer-integrated manufacturing scenarios [4–7]. Origi- nallydevelopedfollowingacrossed-array,theRPDmethodology remainscontroversialdueprimarilytoitsvariousmathematical flawsand statisticalinconsistencies, suchas thecrossed-array’s inability to assess the interaction between control and noise variables [4,7,8]. To resolve suchissues [9,10], proposed using responsesurfacemethodology(RSM)withcombinedarrays.This experimental strategy allows the computation of noise-control interactionsusingacentralcompositedesign(CCD)withembed- dednoisefactors,generatingthemeanandvarianceequationas fromthepropagationoferrorprinciple.

ThegeneralschemeofanRPD-RSMproblemconsistsofper- forminganexperimentaldesignwhileconsideringthenoisefactors tobecontrolvariablesandeliminatingfromthedesignanyaxial pointsrelatedtothenoisefactors[11].Thenapolynomialsurface forf(x,z)isestimatedusingtheOLSorWLSalgorithm,obtaining f(x,z)partialderivatives.Thisprocedureleadstoaresponsesurface forthemean ˆy(x)andanotherforthevarianceˆ²(x),considering thenoise-controlfactorsinteractions.Thisapproachiscalleddual responsesurface(DRS).

http://dx.doi.org/10.1016/j.precisioneng.2014.02.013 0141-6359/©2014ElsevierInc.Allrightsreserved.

f1(x) f2(x)

Pareto Frontier D Utopia Line CHIM

NBI points

f1*(x1*) f1(x2*)

f2(x1*)

f2*(x2*) fU a fN

b c

d e

Anchor point

Anchor point

Fig.1.GraphicaldescriptionofNBImethod.

Applied widely by modern industries, RPD approaches for multiresponse optimization problems have been only sparsely developed[7,12,13].Eveninthoseworksinvolvingmultiresponse approaches, researchersappearto havegenerally neglectedthe noise-controlinteractions,computingthemeanandvarianceequa- tionsfromcrossedarraysordesignreplicates[4,13–19].

IntheDRSmethod,themean ˆy(x)andvarianceˆ²(x)maybe optimizedsimultaneouslyconsideringdifferentschemes[9,12,20], for example, established an optimization scheme considering Minx∈˝ˆ²(x),subjecttotheconstraintof ˆy(x)=T,whereTisthetarget for ˆy(x),andthat,usingaLagrangeanmultiplierapproach,evalu- atesonlyonequalitycharacteristic.[21]presentedabias-specified robust design method formulating a nonlinear optimization programthatminimizesprocessvariabilitysubjecttocustomer- specifiedconstraintsontheprocessbias,suchas|y(x)ˆ −T|≤.The mean,variance,andtargetcanalsobecombinedinamean-squared error(MSE)functionwhichmustbeminimizedandsubjectedtoa setofconstraints,as,forexample,theexperimentalregion.This figurecanbestatedasMin

x∈˝[ˆy(x)−T]²+²[4,12–14,17,22–24].

Supposing that mean and variance may assume differ- ent degrees of importance, the MSE objective function can also be weighted, as MSEw=w₁·(ˆy(x)−T)²+w₂˙cˆ²(x), where the weights w1 and w2 are pre-specified positive constants [10,12,19,24]. Still, these weights can be experimented with throughdifferentconvexcombinations,i.e.,w₁+w₂=1,withw₁>0 andw2>0,generatingasetofnon-inferiorsolutionsformultiple objectiveoptimization[19].

Extending the MSE criterion to multiobjective problems, an operator like a weighted sum may be used [25,26] leading to anobjectivefunctionasMSET=

p

i=1[(ˆy_i−T_i)²+ˆ_i²].Ifdifferent degreesofimportanceareattributedtoeachMSE_i,theglobalobjec- tivefunctioncanbewrittenasproposedby[27]

MSET=

p i=1

w_i·MSE_i=

p i=1

w_i·[(ˆy_i−T_i)²+ˆ²_i] (1)

A common concern with multiobjective MSE optimization is related to the convexity of Pareto frontiers generated using weighted sums. According to [4], in most RPD applications, a second-orderpolynomialmodelisadequatetoaccommodatethe curvatureof processmeanand variance functions.Thus, mean- squaredrobustdesignmodelswouldcontainfourth-orderterms.

Consequently,theassociatedParetofrontiermightbenon-convex and non-supported efficient solutions couldbe generated. It is importanttostatethatadecisionvectorx*∈SisParetooptimalif

theredoesnotexistanotherx∈Ssuchthatf_i(x)≤f_i(x*)foralli=1, 2,...,k.Accordingto[4],forthebi-objectivecase,theweighted sumcanbewrittenasaconvexcombinationoftwoMSEfunctions, suchas:

Min MSET=wMSE₁+(1−w)MSE₂ S.t.: x∈˝ (2) Theweightedsummethod,asdescribedinEq.(2),is widely employedtogeneratethetrade-offsolutionsfornonlinearmulti- objectiveoptimizationproblems.Accordingto[4],thebi-objective problemofEq.(2)isconvexifthefeasiblesetXisconvexandthe MSEfunctionsarealsoconvex.Whenatleastoneobjectivefunction isnotconvex,thebi-objectiveproblembecomesnon-convex,gen- eratinganon-convexandevenunconnectedParetofrontier.The principalconsequenceofanon-convexParetofrontieristhatpoints ontheconcavepartsofthetrade-offsurfacewillnotbeestimated.

Thisinstabilityisduetothefactthattheweightedsumisnota Lipshitzianfunctionoftheweightw[28].Anotherdrawbacktothe weightedsumsisrelatedtotheuniformspreadofPareto-optimal solutions.Evenifauniformspreadofweightvectorsareused,the Paretofrontierwillnotbeequispacedorevenlydistributed[28,29].

Toovercomethesedisadvantages [30],proposedthenormal boundaryintersectionmethod(NBI),showingthattheParetosur- facewillbeevenlydistributedindependentoftherelativescalesof theobjectivefunctions.So,followingtheaforementioneddiscus- sion,thisarticlewillpresentatwo-foldedapproachtocouplingthe NBImethodwithMSEobjectivefunctions.

This paper is organized as follows: Section 2 presents the main characteristics of normal boundary intersection method, discussingtheconceptsofutopialine,payoffmatrixandanchor- age points.Section 3 presents the NBI-MSEmethod; Section 4 presentsanumericalapplicationtoillustratetheadequacyofthe work’sproposal;andalsotheconfirmationrunsthatwerecarried out,demonstratingthemathematicalresultscanbeconfirmedin practice.Section5presentstheresultsanddiscussion.

2. Normalboundaryintersection(NBI)

TheNBImethodshowninFig.1isanoptimizationroutinedevel- oped to finda uniformly spread Pareto-optimal solutions for a generalnon-linearmultiobjectiveproblem[29,30].

ThefirststepintheNBImethodestablishesthepayoffmatrix˚, basedonthecalculationoftheindividualminimaofeachobjective function.Thesolutionthatminimizesthei-thobjectivefunction f_i(x)canberepresentedasf_i^∗(x^∗_i).Whentheindividualoptimax^∗_iis replacedintheremainingobjectivefunctions,fi(x^∗_i)isobtained.In

matrixnotation,thepayoffmatrix˚canbewrittenas:

˚=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

f₁^∗(x₁^∗) ··· f1(x^∗_i) ··· f1(x^∗_m) ..

. . .. ...

fi(x₁^∗) ··· f_i^∗(x^∗_i) ··· f_i^∗(x^∗_m) ..

. . .. ...

fm(x^∗₁) ··· fm(x^∗_i) ··· f_m^∗(x^∗_m)

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

⇒˚¯

=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

f¯₁ ··· f¯₁ ··· f¯₁(x_m^∗) ..

. . .. ...

f¯i ··· f¯i ··· f¯i(x^∗_m) ..

. . .. ...

f¯m(x^∗₁) ··· f¯m(x^∗_i) ··· ¯fm(x^∗_m)

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

(3)

Eachrowof payoff matrix ˚is composedof minimum and maximumvaluesofthei-thobjectivefunctionf_i(x).Thesevalues canbeusedtonormalizetheobjective functions,mainlywhen theywillbe writtenin terms of differentscalesor units.Like- wise,writingavectorwiththesetofindividualminimumf^U= [f₁^∗(x^∗₁)...,f_i^∗(x^∗_i)...,f_m^∗(x^∗_m)]^T,itisobtainedtheUtopiapoint.Analo- gously,byjoiningthemaximumvaluesofeachobjectivefunction f^N=[f₁^N...,f_i^N...,f_m^N]^T,a setcalledtheNadirpoint is obtained.

Thenormalizationoftheobjectivefunctionscanbeobtainedusing thesetwosets,suchas:

¯f(x)= f_i(x)−f_i^U

f_i^N−f_i^U , i=1,...,m (4)

Thisnormalizationleadstothenormalizedpayoffmatrix ¯˚.

Accordingto[28],theconvexcombinationsofeachrowofpay- offmatrixformstheconvexhullofindividualminima(CHIM).An evendisplacementofanypointMalongtheUtopialineleadsaway fromagooddistributionoftheParetopoints.Theanchorpointcor- respondstothesolutionofthesingle-optimizationproblemf_i^x(x^∗_i) [31,32].Thetwoanchorpointsareconnectedby“Utopialine”.This lineisequallydividedinproportionalsegmentslikea,bandein Fig.1.Then,consideringaconvexweightingw,suchas˚w_i,apoint intheCHIMisrepresented.Let ˆndenotethenormalunitdirection (acolumnvectorofones)totheCHIMatthepoint˚w_itowardthe origin;then˚w+Dˆn,withD∈R,representsthesetofpointson thatnormalvector[29,32].TheiterativelymaximizationofDleads toequispacedpointsontheParetoFrontier(Fig.1).

Thepointofintersectionofthenormalandtheboundaryoffea- sibleregionclosesttotheorigincorrespondstothemaximization ofthedistancebetweentheUtopialineandtheParetofrontier.The optimizationproblemcanthenbewrittenas:

Max(x,t) D

subject to: ˚w¯ +Dˆn=F(x)¯

x∈˝

(5)

Thisoptimizationproblemcanbeiterativelysolvedfordiffer- entvaluesofw,creatinganevenlydistributedParetofrontier.A commonchoiceforwissuggestedby[32]aswn=1−

i=1wi.

Theconceptual parameterDcan bealgebraicallyeliminated fromEq.(5).Forbi-dimensionalproblems,forexample,thisexpres- sioncanbesimplifiedas:

Min f¯₁(x)

s.t.: f¯₁(x)−f¯₂(x)+2w−1=0 gj(x)≥0

0≤w≤1

(6)

3. NBIapproachtomultiresponserobustparameter optimizationforcombinedarrays

Accordingto[11],Taguchiproposedthatforrobustoptimiza- tionitcouldbereasonabletosummarizethedatafromacrossed arrayexperimentwiththemeanofeachobservationintheinner arrayacrossallrunsin theouterarray.Thiswasdefinedasthe signal-to-noiseratio.However,Montgomeryemphasizedthatone cannotestimateinteractionsbetweencontrolandnoiseparame- ters,sincesamplemeansandvariancesarecomputedoverthesame levelsofthenoisevariablesinacrossedarraystructure.Interac- tionsamongcontrollableandnoisefactorsare,therefore,thekey tosolvingrobustdesign problems.Thegeneralresponsesurface modelinvolvingcontrolandnoisevariables,organizedinacom- binedarray,maybewrittenas:

y(x,z)=ˇ₀+

k i=1

ˇ_ix_i+

k i=1

ˇ_iix²_i +

i<j

ˇ_ijx_ix_j+

k i=1

_iz_i

+

k i=1

r j=1

ı_ijx_iz_j+ (7)

Assumeindependentnoisevariableswithzeromeanandvari- ances _z². Furthermore, consider that noise variables and the randomerrorareuncorrelated.Withtheseassumptions,themean andvariancemodelscanbewrittenas:

Ez[y(x,z)]=f(x) (8)

Vz[y(x,z)]=_z²_i

_r

i=1

∂y(x,z)

∂zi

2

+² (9)

wherekandrarethenumberofcontrolandnoisevariables,respec- tively.InEq.(9),_z²_i isgenerallyassumedas1and ² iswithin variationobtainedinANOVAanalysisofthefullquadraticmodel of ˆy(x,z).

Inthecontextofrobustparameterdesign,accordingto[5],rel- ativelysignificantbiasinthemean andvariance responsescan resultfromincorrectestimation methodoftheircoefficients. In thisexample,becausethereis onlyonenoisefactor,thesearch areaissmall(2×2)andafullfactorialdesignisused.Usingincor- rectestimationleadstoslightdifferencesbetweentheIMandWLS methods,butinthepresenceofmorenoisefactorsitisexpected morevariation.Hence,oneshouldbemorecautiousaboutthepres- enceofnoisefactorswhenestimatingcoefficientsoftheresponse function.

Let’stakef_i(x)=MSE_i(x)todevelop anNBIapproachtomul- tiresponserobust parameter optimization.Taking f_i^U=MSE^I_i(x),

f_i^N=MSE_i^max(x)andadoptingthescalarizationdescribedbyEq.(4), abidimensionalNBIapproachforMSEfunctionscanbewrittenas:

Min f¯₁(x)=

MSE₁(x)−MSE₁^I(x) MSE^max₁ (x)−MSE₁^I(x)

S.t.: g1(x)=

MSE1(x)−MSE^I₁(x) MSE^max₁ (x)−MSE^I₁(x)

−

MSE2(x)−MSE^I₂(x) MSE^max₂ (x)−MSE₂^I(x)

+2w−1=0 g₂(x)=x^Tx≤²

0≤w≤1

(10)

with:MSE_i(x)=(ˆy_i(x)−T_i)²+²_i(x) (11)

yˆi(x)=Ez[y(x,z)] and _i²(x)=_z²_i

_r

i=1

∂y(x,z)

∂zi

2

+²

Inthisformulation,MSE^I_i(x)correspondstotheindividualopti- mizationofeach MSE_i(x)(utopiavalue),constrainedonlytothe experimentalregion.The denominatorinEq. (10),MSE^max_i (x)− MSE^I_i(x),standsforthenormalizationofmultipleresponses,doing MSE^max_i (x)asthemaximumvalueofpayoffmatrix(matrixformed byallsolutionsobservedintheindividualoptimizations).Thesetof constraintsgj(x)≥0canrepresentanydesiredrestriction,butitis generallyusedtodesignatetheexperimentalregion.Itisclearthat intermsofdesignfactorsthisproposalestablishestheempirical modelsforthemean,variance,andcovariance.Thisiscommonly doneusingcrossedarrays.

Sincetheglobalmultiobjectivefunctionisestablished,itsopti- mumcangenerallybereachedbyusingseveralmethodsavailable tosolvenonlinearprogrammingproblems(NLP),suchthegeneral- izedreducedgradient(GRG)[13,19,33–35].GRGisconsideredone ofthemostrobustandefficientgradientalgorithmsfornonlinear optimizationanditexhibits,asanattractivefeature,anadequate globalconvergence,mainlywheninitiatedsufficientlyclosetothe solution[36].Moreover,onecanseethatthetransformedmultiob- jectivefunctionremainsconvex,sothatastrictminimumshould exist.ItwasforthisreasonthattheGRGwasusedinthisstudy.

TheNBI-MSEapproachproposedinthissectionmaybesumma- rizedintheproposedprocedure’sfollowingsteps:

Step1:Screeningruns

Consideringthecuttingspeedandfeedraterecommendedby thetool manufacturer, several experiments wereconducted as screeningruns.Threerefrigeration conditionsweretestedusing minimum(150ml/min)andmaximum(20l/min)quantityoffluid asalsoadrymillingcondition,adoptingasendoftoollifetheflank toolwearofVBmax=0.2mm.

Step2:Experimentaldesign

Establish an adequate combined array as an experimental design,includingasmuchcontrolandnoisevariablesasdesired.

Runtheexperimentsinrandomorderandstoretheresponses.

Step3:Modelingofresponsesincludingcontrolandnoise variables

Establishequationsfory(x,z)usingexperimentaldatafororig- inalresponses.

Step4:Meansandvariancesdefinition

Establishequationsformeanandvarianceofy(x,z)usingEqs.

(8)and(9).IfthevalueofR²adj.isnotadequate,employtheWLS method,usingasweightstheinverseofquadraticresiduals.

Step5:ConstrainedoptimizationofYp

TheNBI-MSEapproachproposedinthissectionmaybesumma- rizedintheproposedprocedure’sfollowingsteps:

Fig.2. (a)Endmillingprocess;(b)endmillingtool;(c)surfaceroughnessmeasure.

Establish the response targets(T_i)using theindividual con- strained minimization of each response surface, such as Y_p= Minx∈˝[ˆyi(x)].

Step6:Payoffmatrixcalculation

Usingthemean,variance,andtargets,buildseachMSEfunction.

Afterwards,runtheindividualoptimizationofeachMSEfunction suchasMin

x∈˝ [ˆyi(x)−T]²+²,composingtheMSEPayoffmatrix.

ForaBi-objectivecase,itissuggested:

˚=

MSE^I₁(x) MSE^max₁ (x) MSE^max₂ (x) MSE^I₂(x)

(12)

Step7:Scalarization

WiththevaluesofthePayoffmatrix,thescalarizationofMSE functionsispromoted.Forthebivariatecase,itisobtained:

f¯(x)= f_i(x)−f_i^I f_i^Max−f_i^I ⇒

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

¯f1(x)=MSE¯ 1(x)= MSE1(x)−MSE₁^I MSE^Max₁ −MSE^I₁

¯f₂(x)=MSE¯ ₂(x)= MSE₂(x)−MSE₂^I MSE^Max₂ −MSE^I₂ f¯(x)=f_i(x)−f_i^U

f_i^N−f_i^U ⇒

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

f¯1(x)=MSE¯ 1(x)=MSE1(x)−MSE^I₁ MSE^Max₁ −MSE₁^I f¯2(x)=MSE¯ 2(x)=MSE₂(x)−MSE^I₂ MSE^Max₂ −MSE₂^I

(13)

Step8:Runthemultiobjectivenonlinearoptimizationalgo- rithm

Chooseadesiredvalueforω,generallyusingtherange[0;1]iter- atively.Foreachchosenweight,solvethesystemofEq.(11)using the generalized reduced gradient (GRG) algorithm, constrained onlytotheexperimentalregion.

Thenextsectionpresentsanumericalapplicationofthepro- posedapproachusingtheendmillingmachiningprocessofAISI 1045steel. Thenumerical resultsserve tochecktheproposal’s adequacy.

4. TheNBI-RPDoptimizationofendmillingprocess 4.1. Experimentalsetup

Toachievethispaper’sobjective,asetof82experimentswere carried out in a finishing end milling operation of AISI 1045 steel(Fig.2a).Thetoolusedwasapositiveendmill,codeR390- 025A25-11Mwitha 25mmdiameter,enteringangleof r=90^◦,

Fig.3.(a)Newtool;(b)worntool(VBmax=0.30mm).

andamediumstepwiththreeinserts.Threerectangularinserts wereused(Fig.2b)withedgelengthsof11mmeach,codeR390- 11T308M-PMGC1025(Sandvik-Coromant).Thetoolmaterialused wascementedcarbideISOP10coatedwithTiCNandTiNbythe PVDprocess.Thecoatinghardnesswasaround3000HV3andthe substratehardness1650HV3withagrainsizesmallerthan1␮m.

TheworkpiecematerialwasAISI1045steelwithahardnessof approximately180HB.Theworkpiecedimensionswererectangu- larblockswithsquaresectionsof100mm×100mmandlengthsof 300mm.AllthemillingexperimentswerecarriedoutinaFADAL verticalmachiningcenter,modelVMC15,withmaximumspindle rotationof7500RPMand15kWofpowerinthemainmotor.The tooloverhangwas60mm.Thecuttingfluidusedintheexperiments wassyntheticoilQuimaticMEII.

The levels for control and noise factors are described in Tables1 and2, respectively.Thedifferentnoise conditionsfur- nishedbyacombinationoffactorsandlevelsdescribedinTable2 express,insomesense,thepossiblevariationthatcanoccurdur- ingtheend millingoperation, suchasthetool flankwear(z1), thevariationsoncuttingfluidconcentration(z2),andthevaria- tionofcuttingfluidflowrate(z₃).Thesurfaceroughnessvalues areexpectedtosuffersomekindof variationdue totheaction ofthecombined noisefactors. Therefore,the mainobjective of therobustparameterdesignistodeterminethesetupofcontrol parameterscapableofachievingareducedsurfaceroughnesswith minimalvariance,mitigatingtheinfluenceofnoisefactorsonthe processperformance.Measurementsofthetoolflankwear(VB_max) (z₁)werecapturedwithanopticalmicroscope(magnification45×) withimagesacquired bya coupleddigital camera.The criteria adoptedastheendoftoollifewasaflankwearofapproximately VBmax=0.30mmasshowninFig.3(aandb).

Theresponsesmeasuredintheend millingprocesswereRa (thearithmeticaveragesurfaceroughness)andRt(themaximum roughnessheight−distancefromhighestpeaktolowestvalley).In thiswork,bothsurfaceroughnessmetricswereassessedusinga Mitutoyoportableroughnesschecker,modelSurftestSJ201,with acut-offlengthof0.25mm(Fig.4).

4.2. Resultsanddiscussionoftheproposedprocedure

TheNBI-MSEapproachdescribedintheprevioussectionishere discussed.Theresultsofallstepsarepresented.Step1:Screening runs

Preliminaryscreening runsrevealeda behavior oftool flank wearasfunctionofthreecoolantconditions,consideringacutting speedofvc=325m/min,feedrateoffz=0.10/toothanddepthof cutap=1mm,respectively.Fig.5showsmaineffectsoftoolwear accordingtodifferentamounts ofcoolant. Observingthefigure itispossibletoconcludethattheminimumquantityofcoolant isthemoreexperimentalcondition.Withmaximumamountsof

Fig.4. Mitutoyoportableroughnesschecker,modelSurftestSJ201.

fluid,thetoolwearincreasedwhencomparedwithdryandmin- imumquantityofcoolant.Thevaluesofwearareassociatedtoa majorheatshockoccurredintheschemeofintermittentcut.For dryconditions,however,thetoolwearbehaviorisuniform,with increasingvaluesforwearalongofthecuttingtime.Toassessthe toolwearasfunctionofcoolantschemes,ananalysisofcovariance (ANCOVA)weredoneusingthecuttingtimetoachievetheVBmax

asacovariate.

Step2:Experimentaldesign

AssuggestedbyMontogmery[11],acombinedarray(usingcen- tralcompositedesign)fork=7variables(x1,x2,x3,x4,z1,z2andz3) with10centerpointswerecreated,alsodeletingtheaxialpoints relatedtothenoisevariables.Thisprocedureresultedin82exper- iments,described inTables3and 4.Thetwosurfaceroughness metricsweremeasuredthreetimesateachofthreepositionson theworkpiece,computedafterdeterminingthemeanofthenine measurements.

Step3:Modelingofresponsesincludingcontrolandnoise variables

ApplyingtheWLSmethodtoestimatethecoefficientsofthe responsesurfacesforRaandRtprovidesthefollowing:

Ra(x,z)=0.689+0.898x₁+0.041x₂−0.066x₃−0.004x₄ +0.012z1+0.002z2+0.005z3+0.493x²₁+0.096x²₂ +0.010x²₃+0.064x²₄+0.074x1x2−0.087x1x3

+0.030x1x4+0.048x1z1−0.086x1z2+0.042x1z3−0.039x2x3

+0.018x₂x₄+0.013x₂z₁−0.073x₂z₂

−0.012x2z3+0.043x3x4+0.020x3z1−0.034x3z2

+−0.041x3z3−0.052x4z1−0.013x4z2−0.025x4z3 (14)

Rt(x,z)=4.719+3.170x1+0.251x2−0.261x3+0.046x4

+0.877z₁+0.040z₂−0.049z₃+1.039x²₁+0.176x²₂+0.173x²₄ +0.498x₁x₂−0.225x₁x₃+0.233x₁x₄+0.310x₁z₁−0.291x₂z₂ +0.188x1z3−0.020x2x3+0.164x2x4−0.087x2z1+0.210x2z2

−0.127x₂z₃+0.181x₃x₄+0.128x₃z₁−0.109x₃z₂+0.042x₃z₃ +0.158x4z1−0.016x4z2−0.157x4z3 (15)

Table1

Controlfactorsandrespectivelevels.

Parameters Unit Symbol Levels

−2.828 −1.000 0.000 +1.000 +2.828

Feedrate mm/tooth fz 0.01 0.10 0.15 0.20 0.29

Axialdepthofcut mm ap 0.064 0.750 1.125 1.500 2.186

Cuttingspeed m/min Vc 254 300 325 350 396

Radialdepthofcut mm ae 12.26 15.00 16.50 18.00 20.74

Table2

Noisefactorsandrespectivelevels.

Noisefactors Unit Symbol Levels

−1.000 0.000 +1.000

Toolflankwear mm Z1 0.00 0.15 0.30

Cuttingfluidconcentration % Z2 5 10 15

Cuttingfluidflowrate l/min Z3 0 10 20

Fig.5. Maineffectsplotforcuttingtimeversuslubricationtype.

Step4:Meansandvariancesdefinition

Employingthepropagationof errorprinciple andtakingthe partialderivativesofEqs.(14)and(15)therespectivemeansand variancesequationscanbewrittenas:

Ez[Ra(x,z)]=0.689+0.898x1+0.041x2−0.066x3−0.004x4

+0.493x²₁+0.069x²₂+0.010x₃²+0.064x²₄ +0.074x1x2+0.087x1x3+0.030x1x4−0.039x2x3

+0.018x2x4+0.043x3x4 (16)

²[Ra(x)]=(0.1023+0.0477x₁+0.0128x₂+0.0198x₃−0.522x₄)²+(0.0017−0.858x₁−0.0732x₂−0.0335x₃−0.0134x₄)² +(0.0048+0.0423x1−0.0123x2−0.0410x3−0.0254x4)²+

0.90

MSE

(17) Ez[Rt(x,z)]=4.719+3.170x₁+0.251x₂−0.261x₃+0.046x₄+1.039x²₁+0.176x₂²+0.173x²₄+0.498x₁x₂−0.225x₁x₃+0.233x₁x₄

−0.020x2x3+0.164x2x4+0.181x3x4 (18)

²[Rt(x)]=(0.8771+0.3105x₁−0870x₂+0.1284x₃−0.1578x₄)²+(0.0403−0.2909x₁−0.2102x₂−0.1092x₃−0.0164x₄)² +(−0.0492+0.1879x₁−0.1268x₂−0.0419x₃−0.1573x₄)²+

0.90

MSE(Rt)

(19)

AccordingtothediscussionofSection3,themeanandvariance modelsdevelopedusingthecombinedarrayarewrittenintermsof onlycontrolvariables,althoughthenoisefactorswereusedduring theexperimentation.However,giventhatthevarianceequation takes the noise influence into account, the adjustment of the

Table3

Experimentaldesign(PartI).

Run x1 x2 x3 x4 z1 z2 z3 Ra Rt

1 0.10 0.75 300.00 15.00 0.00 5.00 20.00 0.297 2.097

2 0.20 0.75 300.00 15.00 0.00 5.00 0.00 1.807 7.587

3 0.10 1.50 300.00 15.00 0.00 5.00 0.00 0.657 3.467

4 0.20 1.50 300.00 15.00 0.00 5.00 20.00 2.573 8.957

5 0.10 0.75 350.00 15.00 0.00 5.00 0.00 0.353 2.160

6 0.20 0.75 350.00 15.00 0.00 5.00 20.00 3.013 9.327

7 0.10 1.50 350.00 15.00 0.00 5.00 20.00 0.270 1.973

8 0.20 1.50 350.00 15.00 0.00 5.00 0.00 2.417 8.743

9 0.10 0.75 300.00 18.00 0.00 5.00 0.00 0.320 2.087

10 0.20 0.75 300.00 18.00 0.00 5.00 20.00 3.170 11.583

11 0.10 1.50 300.00 18.00 0.00 5.00 20.00 0.280 1.690

12 0.20 1.50 300.00 18.00 0.00 5.00 0.00 2.877 10.187

13 0.10 0.75 350.00 18.00 0.00 5.00 20.00 0.270 2.027

14 0.20 0.75 350.00 18.00 0.00 5.00 0.00 3.030 11.197

15 0.10 1.50 350.00 18.00 0.00 5.00 0.00 0.550 3.340

16 0.20 1.50 350.00 18.00 0.00 5.00 20.00 1.520 7.043

17 0.10 0.75 300.00 15.00 0.30 5.00 0.00 0.497 4.560

18 0.20 0.75 300.00 15.00 0.30 5.00 20.00 2.770 10.973

19 0.10 1.50 300.00 15.00 0.30 5.00 20.00 0.383 2.707

20 0.20 1.50 300.00 15.00 0.30 5.00 0.00 3.247 12.473

21 0.10 0.75 350.00 15.00 0.30 5.00 20.00 0.760 4.647

22 0.20 0.75 350.00 15.00 0.30 5.00 0.00 0.800 4.580

23 0.10 1.50 350.00 15.00 0.30 5.00 0.00 0.500 3.660

24 0.20 1.50 350.00 15.00 0.30 5.00 20.00 2.503 10.757

25 0.10 0.75 300.00 18.00 0.30 5.00 20.00 0.397 2.877

26 0.20 0.75 300.00 18.00 0.30 5.00 0.00 1.063 6.007

27 0.10 1.50 300.00 18.00 0.30 5.00 0.00 0.367 2.007

28 0.20 1.50 300.00 18.00 0.30 5.00 20.00 2.783 15.330

29 0.10 0.75 350.00 18.00 0.30 5.00 0.00 0.763 4.217

30 0.20 0.75 350.00 18.00 0.30 5.00 20.00 1.437 7.253

31 0.10 1.50 350.00 18.00 0.30 5.00 20.00 0.383 3.137

32 0.20 1.50 350.00 18.00 0.30 5.00 0.00 2.960 11.610

33 0.10 0.75 300.00 15.00 0.00 15.00 0.00 0.803 4.007

34 0.20 0.75 300.00 15.00 0.00 15.00 20.00 2.030 7.213

35 0.10 1.50 300.00 15.00 0.00 15.00 20.00 0.537 4.583

36 0.20 1.50 300.00 15.00 0.00 15.00 0.00 2.110 9.117

37 0.10 0.75 350.00 15.00 0.00 15.00 20.00 0.920 4.480

38 0.20 0.75 350.00 15.00 0.00 15.00 0.00 1.743 7.157

39 0.10 1.50 350.00 15.00 0.00 15.00 0.00 0.290 2.043

40 0.20 1.50 350.00 15.00 0.00 15.00 20.00 0.943 4.460

41 0.10 0.75 300.00 18.00 0.00 15.00 20.00 0.513 2.973

controlfactorsleadstotheminimizationoftheprocessvariability, warrantingtherobustnessoftheendmillingprocess.

Fig.6showstheresponsesurfacesforRameanandFig.7shows theresponsesurfacesforRavariance.Ascanbenoted,meanand variancesurfacespresentaminimum,whichsuggeststhattheopti- mizationalgorithmcansearchandfindaglobaloptimumforthe dualmean–variance.ForRttheresultsaresimilar.

Step5:ConstrainedoptimizationofYp

Sincethemeanandvarianceequationsofthetworesponses ofinterest are estimated, theproposedoptimizationprocedure canberun.AnindividualoptimizationofEz[Ra(x,z)]andEz[Rt(x, z)]is conducted, obtainingasthe respectiveoptimathevalues _Ra=0.231␮mand_Rt=1.7954␮m. Thesevalues willbeconsid- eredasthetargetsandwillbeusedinordertocomposeeachMSE(x) function.

Step6:Payoffmatrixcalculation

After individual optimization, one can obtain the values of MSE^max_i (x)andMSE_i^I(x)forbothRaandRt.Forbothcases,theutopia pointsleadtothePayoffmatrixofTable5.

Steps7 and8: Scalarizationandmultiobjective nonlinear optimizationalgorithm

TheresultsofTable6areobtainedbyapplyingtheNBI-MSE methodand doing successiveoptimizations iteratively. For the sakeofcomparison,thesameprocedurewasrepeated,usingthe weightedsummethod.TheresultsaredescribedinTable7.

SuchdatawentintothebuildingoftheParetofrontierswith NBI-MSEmethod(Fig.8)andweightedsums(Fig.9).

It is noteworthy that the NBI-MSE method outperforms the weighted sums as an equispaced frontier, avoiding the agglomeration of optimum points in a portion of extreme curvature in the solution space. It can be observed that in regions where the weighted summethod is incapable of find- ing feasible solutions—creating a discontinuity—the NBI-MSE method generates a good deal of equispaced points. This can occur because the problem is non-convex bi-objective, with at least one MSE function non-convex. Note that the weightedsumsmethodfailstoidentifynon-supportedefficient solutions between the two anchorage points (individual opti- mization), forming a cluster of non-dominated solutions for 0.91≤MSE₁≤0.92 and 1.24≤MSE₂≤1.45. The method mainly fails in the transition from the individual optimization for the first or last weight applied, respectively, w₂=0.05 and w₂₅=0.95. Even though what is considered here is the con- vexportionof thefrontierobtainedbyweightedsums (Fig.7), the solutions, it may be observed, are not evenly distributed along the frontier. Note that in this case it was used incre- ments of approximately 5% in the composition of the fron- tier.

Accordingto[21],abi-objectiveoptimizationproblemisconvex ifthefeasiblesetXisconvexandbothobjectivefunctionsarecon- vex.Inthiscase,fortheresultsofNBI-MSEmethod,theParetoset canbeviewedasaconvexcurveinthespaceof².Furthermore,the constraintx^Tx−²≤0isconvexsinceitrepresentsahypersphere ofradius␳.

Table4

Experimentaldesign(PartII).

Run x1 x2 x3 x4 z1 z2 z3 Ra Rt

42 0.20 0.75 300.00 18.00 0.00 15.00 0.00 2.087 7.550

43 0.10 1.50 300.00 18.00 0.00 15.00 0.00 0.430 2.823

44 0.20 1.50 300.00 18.00 0.00 15.00 20.00 2.557 10.570

45 0.10 0.75 350.00 18.00 0.00 15.00 0.00 0.350 2.457

46 0.20 0.75 350.00 18.00 0.00 15.00 20.00 1.700 6.507

47 0.10 1.50 350.00 18.00 0.00 15.00 20.00 0.617 3.057

48 0.20 1.50 350.00 18.00 0.00 15.00 0.00 1.747 8.273

49 0.10 0.75 300.00 15.00 0.30 15.00 20.00 0.823 4.690

50 0.20 0.75 300.00 15.00 0.30 15.00 0.00 3.007 11.787

51 0.10 1.50 300.00 15.00 0.30 15.00 0.00 0.643 5.230

52 0.20 1.50 300.00 15.00 0.30 15.00 20.00 2.937 9.870

53 0.10 0.75 350.00 15.00 0.30 15.00 0.00 0.803 4.997

54 0.20 0.75 350.00 15.00 0.30 15.00 20.00 2.220 9.797

55 0.10 1.50 350.00 15.00 0.30 15.00 20.00 0.463 2.793

56 0.20 1.50 350.00 15.00 0.30 15.00 0.00 2.203 9.823

57 0.10 0.75 300.00 18.00 0.30 15.00 0.00 0.820 5.343

58 0.20 0.75 300.00 18.00 0.30 15.00 20.00 2.547 10.663

59 0.10 1.50 300.00 18.00 0.30 15.00 20.00 0.377 2.560

60 0.20 1.50 300.00 18.00 0.30 15.00 0.00 2.193 8.853

61 0.10 0.75 350.00 18.00 0.30 15.00 20.00 0.637 4.050

62 0.20 0.75 350.00 18.00 0.30 15.00 0.00 2.247 9.590

63 0.10 1.50 350.00 18.00 0.30 15.00 0.00 0.483 3.400

64 0.20 1.50 350.00 18.00 0.30 15.00 20.00 2.887 11.327

65 0.01 1.13 325.00 16.50 0.15 10.00 10.00 0.100 0.820

66 0.29 1.13 325.00 16.50 0.15 10.00 10.00 2.440 10.760

67 0.15 0.06 325.00 16.50 0.15 10.00 10.00 0.350 1.910

68 0.15 2.19 325.00 16.50 0.15 10.00 10.00 1.573 6.817

69 0.15 1.13 254.29 16.50 0.15 10.00 10.00 0.650 5.257

70 0.15 1.13 395.71 16.50 0.15 10.00 10.00 0.440 3.413

71 0.15 1.13 325.00 12.26 0.15 10.00 10.00 0.390 3.383

72 0.15 1.13 325.00 20.74 0.15 10.00 10.00 1.183 6.230

73 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.343 2.990

74 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.540 3.283

75 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.680 4.083

76 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.520 3.247

77 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.540 4.090

78 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.323 2.993

79 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.527 4.990

80 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.607 3.453

81 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.697 4.970

82 0.15 1.13 325.00 16.50 0.15 10.00 10.00 0.430 2.863

Fig.6. ResponsesurfacesforRamean.

Table5 Payoffmatrices.

PayoffmatrixforRaandRt PayoffmatrixforMSE1andMSE2

0.2301 0.4781 0.9079 0.9568

2.3675 1.7954 1.9679 1.2173

4.3. Confirmationruns

In robust design optimization,theidea is tofinda setupof controllablefactorsthatareinsensibletotheactionsoftheuncon- trollableones.Totestthisclaimwiththeprocessunderstudy,a L9Taguchidesignwasusedtoassessthebehavioroftheoptimum setupinarangeofscenariosformedbythenoisefactors.Ifthe

Fig.7.ResponsesurfacesforRavariance.

Table6

OptimizationresultswithNBI-MSEmethod.

Weights x1 x2 x3 x4 Ra Rt VarRa VarRt MSE1 MSE2

0.00 −1.450 0.855 −0.347 1.022 0.446 1.977 0.910 1.184 0.957 1.217

0.05 −1.434 0.842 −0.408 1.033 0.435 1.979 0.910 1.184 0.952 1.218

0.10 −1.418 0.826 −0.472 1.041 0.424 1.981 0.910 1.184 0.947 1.219

0.15 −1.401 0.807 −0.538 1.047 0.411 1.984 0.909 1.186 0.942 1.221

0.20 −1.382 0.784 −0.609 1.051 0.398 1.987 0.909 1.187 0.938 1.224

0.25 −1.363 0.757 −0.684 1.050 0.385 1.991 0.909 1.190 0.933 1.229

0.30 −1.341 0.726 −0.764 1.045 0.370 1.997 0.909 1.195 0.929 1.235

0.35 −1.316 0.689 −0.852 1.033 0.353 2.007 0.909 1.200 0.924 1.245

0.40 −1.286 0.648 −0.949 1.013 0.335 2.022 0.909 1.207 0.920 1.258

0.45 −1.248 0.607 −1.054 0.981 0.317 2.050 0.909 1.215 0.917 1.279

0.50 −1.199 0.574 −1.159 0.943 0.301 2.099 0.909 1.220 0.914 1.312

0.55 −1.146 0.552 −1.224 0.909 0.290 2.168 0.909 1.221 0.912 1.359

0.60 −1.115 0.535 −1.106 0.834 0.287 2.235 0.908 1.222 0.911 1.415

0.65 −1.084 0.516 −1.006 0.778 0.285 2.297 0.907 1.225 0.910 1.476

0.70 −1.055 0.497 −0.920 0.733 0.283 2.354 0.907 1.229 0.909 1.540

0.75 −1.026 0.478 −0.847 0.698 0.282 2.407 0.906 1.233 0.909 1.607

0.80 −0.998 0.458 −0.784 0.669 0.281 2.458 0.906 1.238 0.908 1.677

0.85 −0.972 0.437 −0.731 0.646 0.280 2.505 0.906 1.244 0.908 1.748

0.90 −0.946 0.416 −0.684 0.626 0.280 2.550 0.906 1.250 0.908 1.820

0.95 −0.922 0.395 −0.643 0.609 0.280 2.594 0.905 1.256 0.908 1.893

1.00 −0.899 0.373 −0.607 0.594 0.281 2.635 0.905 1.263 0.908 1.968

Table7

Optimizationresultswithweightedsums.

Weights x1 x2 x3 x4 Ra Rt VarRa VarRt MSE1 MSE2

0.00 −1.450 0.855 −0.347 1.022 0.446 1.977 0.910 1.184 0.957 1.217

0.05 −1.445 0.851 −0.369 1.026 0.442 1.978 0.910 1.184 0.955 1.217

0.10 −1.438 0.846 −0.392 1.030 0.438 1.979 0.910 1.184 0.953 1.218

0.15 −1.432 0.840 −0.417 1.034 0.433 1.979 0.910 1.184 0.951 1.218

0.20 −1.425 0.833 −0.444 1.038 0.429 1.980 0.910 1.184 0.949 1.218

0.25 −1.418 0.826 −0.473 1.041 0.423 1.981 0.910 1.184 0.947 1.219

0.30 −1.410 0.817 −0.503 1.045 0.418 1.982 0.910 1.185 0.945 1.220

0.35 −1.401 0.808 −0.536 1.047 0.412 1.983 0.909 1.185 0.943 1.221

0.40 −1.393 0.797 −0.570 1.049 0.406 1.985 0.909 1.186 0.940 1.222

0.45 −1.383 0.785 −0.606 1.051 0.399 1.987 0.909 1.187 0.938 1.224

0.50 −1.373 0.771 −0.645 1.051 0.392 1.989 0.909 1.189 0.935 1.226

0.55 −1.362 0.756 −0.685 1.050 0.384 1.991 0.909 1.191 0.933 1.229

0.60 −1.351 0.740 −0.728 1.048 0.376 1.994 0.909 1.193 0.931 1.232

0.65 −1.338 0.722 −0.773 1.044 0.368 1.998 0.909 1.195 0.928 1.236

0.70 −1.325 0.702 −0.821 1.038 0.359 2.003 0.909 1.198 0.926 1.241

0.75 −1.309 0.680 −0.873 1.030 0.349 2.009 0.909 1.202 0.923 1.247

0.80 −1.292 0.656 −0.930 1.018 0.339 2.019 0.909 1.206 0.921 1.256

0.85 −1.270 0.630 −0.994 1.001 0.327 2.032 0.909 1.210 0.919 1.267

0.90 −1.242 0.601 −1.070 0.976 0.314 2.055 0.909 1.216 0.916 1.283

0.95 −1.193 0.571 −1.170 0.939 0.299 2.105 0.909 1.221 0.914 1.317

1.00 −0.899 0.373 −0.607 0.594 0.281 2.635 0.905 1.263 0.908 1.968